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Differential inequalities in linear- and affine-invariant families of harmonic mappings
Abstract
In linear- and affine-invariant families of harmonic mappings of the unit disk we prove some differential inequalities such
as the sharp two-sided estimate of the Jacobian and an estimate of the curvature of the image of the circle.
Key words and phrasesharmonic mappings-linear- and affine-invariant families of functions-order of a family
... depend on supremum of the absolute value of the second coefficients a 2 over the family L. The notion of order of family L was defined by T. Sheil-Small [7] as α = sup f ∈L |a 2 | (exactly in the same form as in the analytic case, cf., [8]). This paper of Sheil-Small on linear and affine invariant families of harmonic functions attracted attention of many mathematicians (cf., [9,10,11]). In 2004 V. Starkov ([12], see also [10] for details) gave definition of new order of l.i.f. as ...
... The sharp estimation of the modulus of this ratio in the a.l.i.f. L was obtained in [11] (the proof is available in [15]) and yields the inequality ...
... Now we will use equality (12) together with estimation (13) in order to obtain an upper bound for |a 3 − a 2 2 | in (11): ...
We obtain estimations of the pre-Schwarzian and Schwarzian derivatives in terms of the order of family in linear and affine invariant families L of sense preserving harmonic mappings of the unit disk D. As the converse result the order of family L is estimated in terms of suprema of Schwarzian and pre-Schwarzian norms over the family L. Main results are obtained by means of theory of linear invariant families.
... However, the sharp upper bounds of α 0 have been obtained for harmonic functions with some special geometric properties (cf., [3,5]). For more results on linear-and affine-invariant families of harmonic functions, see [6,11,16,17]. ...
... Proof. Let f satisfy the conditions of Theorem 3. The upper and lower bounds of curvature k f (z) of images f (γ r ) of the circle γ r , r ∈ (0,1), in the linear-and affine-invariant families of harmonic sense-preserving in D functions f were published in [11]. In particular, it was proved that f is convex in the disk |z| < α 0 + β 0 − (α 0 + β 0 ) 2 − 1 and ...
The plane domain D is called R-convex if D contains each compact set bounded by two shortest sub-arcs of the radius R with endpoints w1, w2 ∈ D, |w1−w2
... where Φ(z) = (z + z 0 )/(1 + z 0 z) and z 0 runs over the disk D. Properties of the linear and affine invariant families of harmonic functions can be found in [21,22,12]. Note that the order of an univalent analytic or univalent sense preserving harmonic function is always finite (cf., [6,8]). ...
... Thus,p is also a Nehari function. So, if (12) holds for a function f ρ,ε in |z| ≤ ρ 1 , then forf ρ,ε = f ρ,ε (ρ 1 z) we have ...
In the paper we obtain some analogues of Nehari’s
univalence conditions for sense-preserving functions that are harmonic in the unit disc D = {z ∈ C : |z| < 1}.
... where α = ord (AL H ). Later this result was generalized to the families of locally univalent harmonic mappings [15]. Now we will show the radius of convexity will be altered under the assumption of Q-quasiconformality of functions f . ...
For a locally univalent sense-preserving harmonic mapping
defined on the unit disk \ID =\{z\in\mathbb C:\, |z|<1\}, let be the
radius of the largest (univalent) disk on the manifold f(\ID) centered at
(). One of the aims of the present investigation is to obtain
sharp upper and lower bounds for the quotient , especially,
for a family of locally univalent Q-quasiconformal harmonic mappings
on \ID. In addition to several other consequences of our
investigation, the disk of convexity of functions belonging to certain linear
invariant families of locally univalent Q-quasiconformal harmonic mappings of
order is also established.
... Let α 0 and β 0 be the supremum of |a 2 | and |b 2 |, respectively, among all functions f ∈ L 0 of the form (1) with b 1 = 0. In [7] and [8], the authors studied the classes L and L 0 and derived the following interesting results. ...
In this paper, we first prove the coefficient conjecture of Clunie and
Sheil-Small for a class of univalent harmonic functions which includes
functions convex in some direction. Next, we prove growth and covering theorems
and some related results. Finally, we propose two conjectures. An affirmative
answer to one of which would then imply for example a solution to the
conjecture of Clunie and Sheil-Small.
Let denote the set of all sense-preserving harmonic mappings in the unit disk \ID, normalized with and . In this paper, we investigate some properties of certain subclasses of , including inclusion relations and stability analysis by precise examples, coefficient bounds, growth, covering and distortion theorems. As applications, we build some Bohr inequalities for these subclasses by means of subordination. Among these subclasses, six classes consist of functions such that is univalent (or convex) in \D for each (or for some , or for some ). Simple analysis shows that if the function belongs to a given class from these six classes, then the functions belong to corresponding class for all . We call these classes as stable classes.
В статье анализируются результаты биохимического состава растительного опада различных фитогенных зон Acer negundo L. в условиях нарушенных пойменных растительных сообществ. Отбор образцов проводили на пробных площадках в различных условиях сомкнутости крон с учетом зон влияния деревьев. В качестве контроля выбрана внешняя зона одиночных деревьев. Определение зольности проводили путем сухого озоления; содержание азота и фосфора – из одной навески после мокрого озоления: азот – по методу Къельдаля, фосфор – по методу Мерфи и Райли; накопление лигнина – с использованием 72% раствора серной кислоты. Статистическая обработка полученных данных и построение графиков выполнены с помощью стандартного пакета программ StatSoft STATISTICA 8.0. for Windows и Microsoft Office Excel 2007. Выявлены некоторые особенности химического состава растительного опада A. negundo в условиях нарушенных пойменных сообществ. У одиночных деревьев в несомкнутых древостоях в подкроновой и прикроновой зонах выявлено наибольшее количество золы, а у деревьев с сомкнутостью крон 50-60% – более высокие показатели по азоту, фосфору и лигнину в сравнении с другими группами деревьев и с контролем. Наиболее сильно различавшимся показателем химического состава растительного опада на пробных площадках было содержание золы и лигнина, в меньшей степени варьировало содержание азота и фосфора. Экспериментальные данные можно использовать для оценки состояния напочвенного покрова и образования органического вещества почвы в лесных сообществах.
We give a lower estimate for the Bloch constant for planar har- monic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a spe- cial case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.
1. Preliminaries 2. Local properties of harmonic mappings 3. Harmonic mappings onto convex regions 4. Harmonic self-mappings of the disk 5. Harmonic univalent functions 6. Extremal problems 7. Mapping problems 8. Additional topics 9. Minimal surfaces 10. Curvature of minimal surfaces Appendix References.
A homeomorphism w = f(z) of a domain D is called a locally quasiconformal mapping, if for each subdomain D′ of D with D'̄ ⊂ D, the restriction of f(z) on D′ is a quasiconformal mapping. We give some conditions for a measurable function μ(z) on the unit disc to be the complex dilatation of a locally quasiconformal mapping f which can be homeomorphically extended to the closed unit disc. © 2013 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.